1 | include "ASM/ASMCosts.ma". |
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2 | include "ASM/WellLabeled.ma". |
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3 | include "ASM/Status.ma". |
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4 | include "common/StructuredTraces.ma". |
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5 | include "arithmetics/bigops.ma". |
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6 | include alias "arithmetics/nat.ma". |
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7 | include alias "basics/logic.ma". |
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8 | |
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9 | definition current_instruction0 ≝ |
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10 | λcode_memory: BitVectorTrie Byte 16. |
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11 | λprogram_counter: Word. |
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12 | \fst (\fst (fetch … code_memory program_counter)). |
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13 | |
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14 | definition current_instruction ≝ |
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15 | λs: Status. |
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16 | current_instruction0 (code_memory … s) (program_counter … s). |
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17 | |
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18 | definition ASM_classify0: instruction → status_class ≝ |
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19 | λi: instruction. |
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20 | match i with |
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21 | [ RealInstruction pre ⇒ |
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22 | match pre with |
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23 | [ RET ⇒ cl_return |
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24 | | JZ _ ⇒ cl_jump |
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25 | | JNZ _ ⇒ cl_jump |
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26 | | JC _ ⇒ cl_jump |
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27 | | JNC _ ⇒ cl_jump |
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28 | | JB _ _ ⇒ cl_jump |
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29 | | JNB _ _ ⇒ cl_jump |
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30 | | JBC _ _ ⇒ cl_jump |
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31 | | CJNE _ _ ⇒ cl_jump |
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32 | | DJNZ _ _ ⇒ cl_jump |
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33 | | _ ⇒ cl_other |
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34 | ] |
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35 | | ACALL _ ⇒ cl_call |
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36 | | LCALL _ ⇒ cl_call |
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37 | | JMP _ ⇒ cl_call |
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38 | | AJMP _ ⇒ cl_jump |
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39 | | LJMP _ ⇒ cl_jump |
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40 | | SJMP _ ⇒ cl_jump |
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41 | | _ ⇒ cl_other |
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42 | ]. |
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43 | |
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44 | definition ASM_classify: Status → status_class ≝ |
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45 | λs: Status. |
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46 | ASM_classify0 (current_instruction s). |
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47 | |
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48 | definition current_instruction_is_labelled ≝ |
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49 | λcost_labels: BitVectorTrie costlabel 16. |
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50 | λs: Status. |
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51 | let pc ≝ program_counter … s in |
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52 | match lookup_opt … pc cost_labels with |
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53 | [ None ⇒ False |
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54 | | _ ⇒ True |
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55 | ]. |
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56 | |
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57 | definition next_instruction_properly_relates_program_counters ≝ |
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58 | λbefore: Status. |
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59 | λafter : Status. |
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60 | let size ≝ current_instruction_cost before in |
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61 | let pc_before ≝ program_counter … before in |
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62 | let pc_after ≝ program_counter … after in |
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63 | let sum ≝ \snd (half_add … pc_before (bitvector_of_nat … size)) in |
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64 | sum = pc_after. |
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65 | |
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66 | definition ASM_abstract_status: BitVectorTrie costlabel 16 → abstract_status ≝ |
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67 | λcost_labels. |
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68 | mk_abstract_status |
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69 | Status |
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70 | (λs,s'. (execute_1 s) = s') |
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71 | (λs,class. ASM_classify s = class) |
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72 | (current_instruction_is_labelled cost_labels) |
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73 | next_instruction_properly_relates_program_counters. |
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74 | |
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75 | let rec compute_max_trace_label_label_cost |
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76 | (cost_labels: BitVectorTrie costlabel 16) |
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77 | (trace_ends_flag: trace_ends_with_ret) |
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78 | (start_status: Status) (final_status: Status) |
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79 | (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag |
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80 | start_status final_status) on the_trace: nat ≝ |
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81 | match the_trace with |
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82 | [ tll_base ends_flag initial final given_trace labelled_proof ⇒ |
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83 | compute_max_trace_any_label_cost … given_trace |
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84 | ] |
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85 | and compute_max_trace_any_label_cost |
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86 | (cost_labels: BitVectorTrie costlabel 16) |
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87 | (trace_ends_flag: trace_ends_with_ret) |
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88 | (start_status: Status) (final_status: Status) |
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89 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
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90 | on the_trace: nat ≝ |
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91 | match the_trace with |
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92 | [ tal_base_not_return the_status _ _ _ _ ⇒ current_instruction_cost the_status |
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93 | | tal_base_return the_status _ _ _ ⇒ current_instruction_cost the_status |
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94 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
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95 | _ _ _ call_trace final_trace ⇒ |
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96 | let current_instruction_cost ≝ current_instruction_cost pre_fun_call in |
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97 | let call_trace_cost ≝ compute_max_trace_label_return_cost … call_trace in |
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98 | let final_trace_cost ≝ compute_max_trace_any_label_cost cost_labels end_flag … final_trace in |
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99 | call_trace_cost + current_instruction_cost + final_trace_cost |
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100 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
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101 | let current_instruction_cost ≝ current_instruction_cost status_pre in |
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102 | let tail_trace_cost ≝ |
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103 | compute_max_trace_any_label_cost cost_labels end_flag |
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104 | status_init status_end tail_trace |
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105 | in |
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106 | current_instruction_cost + tail_trace_cost |
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107 | ] |
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108 | and compute_max_trace_label_return_cost |
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109 | (cost_labels: BitVectorTrie costlabel 16) |
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110 | (start_status: Status) (final_status: Status) |
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111 | (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status) |
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112 | on the_trace: nat ≝ |
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113 | match the_trace with |
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114 | [ tlr_base before after trace_to_lift ⇒ compute_max_trace_label_label_cost … trace_to_lift |
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115 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ |
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116 | let labelled_cost ≝ compute_max_trace_label_label_cost … labelled_trace in |
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117 | let return_cost ≝ compute_max_trace_label_return_cost … ret_trace in |
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118 | labelled_cost + return_cost |
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119 | ]. |
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120 | |
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121 | include alias "arithmetics/nat.ma". |
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122 | |
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123 | let rec compute_max_trace_label_label_cost_is_ok |
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124 | (cost_labels: BitVectorTrie costlabel 16) |
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125 | (trace_ends_flag: trace_ends_with_ret) |
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126 | (start_status: Status) (final_status: Status) |
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127 | (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag |
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128 | start_status final_status) on the_trace: |
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129 | clock … final_status = (compute_max_trace_label_label_cost cost_labels trace_ends_flag start_status final_status the_trace) + (clock … start_status) ≝ ? |
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130 | and compute_max_trace_any_label_cost_is_ok |
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131 | (cost_labels: BitVectorTrie costlabel 16) |
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132 | (trace_ends_flag: trace_ends_with_ret) |
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133 | (start_status: Status) (final_status: Status) |
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134 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
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135 | on the_trace: |
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136 | clock … final_status = (compute_max_trace_any_label_cost cost_labels trace_ends_flag start_status final_status the_trace) + (clock … start_status) ≝ ? |
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137 | and compute_max_trace_label_return_cost_is_ok |
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138 | (cost_labels: BitVectorTrie costlabel 16) |
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139 | (start_status: Status) (final_status: Status) |
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140 | (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status) |
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141 | on the_trace: |
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142 | clock … final_status = (compute_max_trace_label_return_cost cost_labels start_status final_status the_trace) + clock … start_status ≝ ?. |
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143 | [1: |
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144 | cases the_trace |
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145 | #ends_flag #start_status #end_status #any_label_trace #is_costed |
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146 | normalize @compute_max_trace_any_label_cost_is_ok |
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147 | |2: |
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148 | cases the_trace |
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149 | [1,2: |
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150 | #start_status #final_status #is_next #is_not_return try (#is_costed) |
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151 | change with (current_instruction_cost start_status) in ⊢ (???(?%?)); |
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152 | cases(is_next) @execute_1_ok |
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153 | |3: |
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154 | #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call |
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155 | #status_final #is_next #is_call #is_after_return #call_trace #final_trace |
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156 | change with ( |
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157 | let current_instruction_cost ≝ current_instruction_cost status_pre_fun_call in |
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158 | let call_trace_cost ≝ compute_max_trace_label_return_cost … call_trace in |
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159 | let final_trace_cost ≝ compute_max_trace_any_label_cost cost_labels end_flag … final_trace in |
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160 | call_trace_cost + current_instruction_cost + final_trace_cost) in ⊢ (???(?%?)); |
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161 | normalize nodelta; |
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162 | >(compute_max_trace_any_label_cost_is_ok cost_labels end_flag status_after_fun_call |
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163 | status_final final_trace) |
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164 | >(compute_max_trace_label_return_cost_is_ok cost_labels status_start_fun_call |
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165 | status_after_fun_call call_trace) |
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166 | cases(is_next) in match (clock … status_start_fun_call); |
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167 | >(execute_1_ok status_pre_fun_call) |
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168 | <associative_plus in ⊢ (??%?); |
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169 | <commutative_plus in match ( |
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170 | compute_max_trace_any_label_cost cost_labels end_flag status_after_fun_call status_final final_trace |
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171 | + compute_max_trace_label_return_cost cost_labels status_start_fun_call status_after_fun_call call_trace); |
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172 | >associative_plus in ⊢ (??%?); |
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173 | <commutative_plus in match ( |
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174 | compute_max_trace_any_label_cost cost_labels end_flag status_after_fun_call status_final final_trace |
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175 | + (current_instruction_cost status_pre_fun_call |
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176 | + clock (BitVectorTrie Byte 16) status_pre_fun_call)); |
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177 | >associative_plus in ⊢ (??%?); |
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178 | <commutative_plus in match ( |
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179 | clock (BitVectorTrie Byte 16) status_pre_fun_call |
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180 | + compute_max_trace_any_label_cost cost_labels end_flag status_after_fun_call status_final final_trace); |
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181 | <(associative_plus ( |
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182 | (compute_max_trace_label_return_cost cost_labels status_start_fun_call status_after_fun_call call_trace))) |
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183 | <associative_plus in ⊢ (??%?); % |
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184 | |4: |
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185 | #end_flag #status_pre #status_init #status_end #is_next |
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186 | #trace_any_label #is_other #is_not_costed |
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187 | change with ( |
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188 | let current_instruction_cost ≝ current_instruction_cost status_pre in |
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189 | let tail_trace_cost ≝ |
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190 | compute_max_trace_any_label_cost cost_labels end_flag |
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191 | status_init status_end trace_any_label |
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192 | in |
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193 | current_instruction_cost + tail_trace_cost) in ⊢ (???(?%?)); |
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194 | normalize nodelta; |
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195 | >(compute_max_trace_any_label_cost_is_ok cost_labels end_flag |
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196 | status_init status_end trace_any_label) |
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197 | cases(is_next) in match (clock … status_init); |
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198 | >(execute_1_ok status_pre) |
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199 | >commutative_plus >associative_plus >associative_plus @eq_f |
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200 | @commutative_plus |
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201 | ] |
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202 | |3: |
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203 | cases the_trace |
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204 | [1: |
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205 | #status_before #status_after #trace_to_lift |
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206 | normalize @compute_max_trace_label_label_cost_is_ok |
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207 | |2: |
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208 | #status_initial #status_labelled #status_final #labelled_trace #ret_trace |
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209 | normalize |
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210 | >(compute_max_trace_label_return_cost_is_ok cost_labels status_labelled status_final ret_trace); |
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211 | >(compute_max_trace_label_label_cost_is_ok cost_labels doesnt_end_with_ret status_initial status_labelled labelled_trace); |
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212 | <associative_plus in ⊢ (??%?); |
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213 | >commutative_plus in match ( |
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214 | compute_max_trace_label_return_cost cost_labels status_labelled status_final ret_trace |
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215 | + compute_max_trace_label_label_cost cost_labels doesnt_end_with_ret status_initial status_labelled labelled_trace); |
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216 | % |
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217 | ] |
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218 | ]. |
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219 | qed. |
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220 | |
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221 | (* XXX: work below here: *) |
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222 | |
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223 | let rec compute_paid_trace_any_label |
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224 | (cost_labels: BitVectorTrie costlabel 16) |
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225 | (trace_ends_flag: trace_ends_with_ret) |
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226 | (start_status: Status) (final_status: Status) |
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227 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
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228 | on the_trace: nat ≝ |
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229 | match the_trace with |
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230 | [ tal_base_not_return the_status _ _ _ _ ⇒ current_instruction_cost the_status |
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231 | | tal_base_return the_status _ _ _ ⇒ current_instruction_cost the_status |
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232 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
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233 | _ _ _ call_trace final_trace ⇒ |
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234 | let current_instruction_cost ≝ current_instruction_cost pre_fun_call in |
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235 | let final_trace_cost ≝ compute_paid_trace_any_label cost_labels end_flag … final_trace in |
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236 | current_instruction_cost + final_trace_cost |
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237 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
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238 | let current_instruction_cost ≝ current_instruction_cost status_pre in |
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239 | let tail_trace_cost ≝ |
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240 | compute_paid_trace_any_label cost_labels end_flag |
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241 | status_init status_end tail_trace |
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242 | in |
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243 | current_instruction_cost + tail_trace_cost |
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244 | ]. |
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245 | |
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246 | definition compute_paid_trace_label_label ≝ |
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247 | λcost_labels: BitVectorTrie costlabel 16. |
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248 | λtrace_ends_flag: trace_ends_with_ret. |
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249 | λstart_status: Status. |
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250 | λfinal_status: Status. |
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251 | λthe_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag |
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252 | start_status final_status. |
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253 | match the_trace with |
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254 | [ tll_base ends_flag initial final given_trace labelled_proof ⇒ |
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255 | compute_paid_trace_any_label … given_trace |
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256 | ]. |
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257 | |
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258 | let rec compute_trace_label_label_cost_using_paid |
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259 | (cost_labels: BitVectorTrie costlabel 16) |
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260 | (trace_ends_flag: trace_ends_with_ret) |
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261 | (start_status: Status) (final_status: Status) |
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262 | (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag |
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263 | start_status final_status) on the_trace: nat ≝ |
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264 | match the_trace with |
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265 | [ tll_base ends_flag initial final given_trace labelled_proof ⇒ |
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266 | compute_paid_trace_label_label cost_labels … the_trace + |
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267 | compute_trace_any_label_cost_using_paid … given_trace |
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268 | ] |
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269 | and compute_trace_any_label_cost_using_paid |
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270 | (cost_labels: BitVectorTrie costlabel 16) |
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271 | (trace_ends_flag: trace_ends_with_ret) |
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272 | (start_status: Status) (final_status: Status) |
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273 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
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274 | on the_trace: nat ≝ |
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275 | match the_trace with |
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276 | [ tal_base_not_return the_status _ _ _ _ ⇒ 0 |
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277 | | tal_base_return the_status _ _ _ ⇒ 0 |
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278 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
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279 | _ _ _ call_trace final_trace ⇒ |
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280 | let call_trace_cost ≝ compute_trace_label_return_cost_using_paid … call_trace in |
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281 | let final_trace_cost ≝ compute_trace_any_label_cost_using_paid cost_labels end_flag … final_trace in |
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282 | call_trace_cost + final_trace_cost |
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283 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
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284 | compute_trace_any_label_cost_using_paid cost_labels end_flag |
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285 | status_init status_end tail_trace |
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286 | ] |
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287 | and compute_trace_label_return_cost_using_paid |
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288 | (cost_labels: BitVectorTrie costlabel 16) |
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289 | (start_status: Status) (final_status: Status) |
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290 | (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status) |
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291 | on the_trace: nat ≝ |
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292 | match the_trace with |
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293 | [ tlr_base before after trace_to_lift ⇒ compute_trace_label_label_cost_using_paid … trace_to_lift |
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294 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ |
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295 | let labelled_cost ≝ compute_trace_label_label_cost_using_paid … labelled_trace in |
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296 | let return_cost ≝ compute_trace_label_return_cost_using_paid … ret_trace in |
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297 | labelled_cost + return_cost |
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298 | ]. |
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299 | |
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300 | let rec compute_trace_label_label_cost_using_paid_ok |
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301 | (cost_labels: BitVectorTrie costlabel 16) |
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302 | (trace_ends_flag: trace_ends_with_ret) |
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303 | (start_status: Status) (final_status: Status) |
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304 | (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag |
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305 | start_status final_status) on the_trace: |
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306 | compute_trace_label_label_cost_using_paid cost_labels … the_trace = |
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307 | compute_max_trace_label_label_cost … the_trace ≝ ? |
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308 | and compute_trace_any_label_cost_using_paid_ok |
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309 | (cost_labels: BitVectorTrie costlabel 16) |
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310 | (trace_ends_flag: trace_ends_with_ret) |
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311 | (start_status: Status) (final_status: Status) |
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312 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) |
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313 | trace_ends_flag start_status final_status) on the_trace: |
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314 | compute_paid_trace_any_label cost_labels trace_ends_flag … the_trace |
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315 | +compute_trace_any_label_cost_using_paid cost_labels trace_ends_flag … the_trace |
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316 | =compute_max_trace_any_label_cost cost_labels trace_ends_flag … the_trace ≝ ? |
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317 | and compute_trace_label_return_cost_using_paid_ok |
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318 | (cost_labels: BitVectorTrie costlabel 16) |
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319 | (start_status: Status) (final_status: Status) |
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320 | (the_trace: trace_label_return (ASM_abstract_status cost_labels) |
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321 | start_status final_status) on the_trace: |
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322 | compute_trace_label_return_cost_using_paid cost_labels … the_trace = |
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323 | compute_max_trace_label_return_cost cost_labels … the_trace ≝ ?. |
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324 | [ cases the_trace #endsf #ss #es #tr #H normalize |
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325 | @compute_trace_any_label_cost_using_paid_ok |
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326 | | cases the_trace |
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327 | [ #ss #fs #H1 #H2 #H3 whd in ⊢ (??(?%%)%); <plus_n_O % |
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328 | | #ss #fs #H1 #H2 whd in ⊢ (??(?%%)%); <plus_n_O % |
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329 | | #ef #spfc #ssfc #safc #sf #H1 #H2 #H3 #tr1 #tr2 whd in ⊢ (??(?%%)%); |
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330 | <compute_trace_any_label_cost_using_paid_ok |
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331 | <compute_trace_label_return_cost_using_paid_ok |
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332 | -compute_trace_label_label_cost_using_paid_ok |
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333 | -compute_trace_label_return_cost_using_paid_ok |
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334 | -compute_trace_any_label_cost_using_paid_ok |
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335 | >commutative_plus in ⊢ (???(?%?)); |
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336 | >commutative_plus in ⊢ (??(??%)?); |
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337 | >associative_plus >associative_plus in ⊢ (???%); @eq_f2 try % |
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338 | <associative_plus <commutative_plus % |
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339 | | #ef #sp #si #se #H1 #tr #H2 #H3 whd in ⊢ (??(?%%)%); >associative_plus @eq_f2 |
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340 | [ % | @compute_trace_any_label_cost_using_paid_ok ] |
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341 | ] |
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342 | | cases the_trace |
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343 | [ #sb #sa #tr normalize @compute_trace_label_label_cost_using_paid_ok |
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344 | | #si #sl #sf #tr1 #tr2 normalize @eq_f2 |
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345 | [ @compute_trace_label_label_cost_using_paid_ok |
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346 | | @compute_trace_label_return_cost_using_paid_ok ]]] |
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347 | qed. |
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348 | |
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349 | (* |
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350 | let rec compute_paid_trace_label_return |
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351 | (cost_labels: BitVectorTrie costlabel 16) |
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352 | (start_status: Status) (final_status: Status) |
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353 | (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status) |
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354 | on the_trace: nat ≝ |
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355 | match the_trace with |
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356 | [ tlr_base before after trace_to_lift ⇒ compute_paid_trace_label_label … trace_to_lift |
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357 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ |
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358 | let labelled_cost ≝ compute_paid_trace_label_label … labelled_trace in |
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359 | let return_cost ≝ compute_paid_trace_label_return … ret_trace in |
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360 | labelled_cost + return_cost |
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361 | ]. |
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362 | *) |
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363 | |
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364 | let rec compute_cost_trace_label_label |
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365 | (cost_labels: BitVectorTrie costlabel 16) |
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366 | (trace_ends_flag: trace_ends_with_ret) |
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367 | (start_status: Status) (final_status: Status) |
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368 | (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag |
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369 | start_status final_status) on the_trace: |
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370 | list (Σk:costlabel. ∃pc. lookup_opt … pc cost_labels = Some … k) ≝ |
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371 | match the_trace with |
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372 | [ tll_base ends_flag initial final given_trace labelled_proof ⇒ |
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373 | let pc ≝ program_counter … initial in |
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374 | let label ≝ |
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375 | match lookup_opt … pc cost_labels return λx. match x with [None ⇒ False | Some _ ⇒ True] → costlabel with |
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376 | [ None ⇒ λabs. ⊥ |
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377 | | Some l ⇒ λ_. l ] labelled_proof in |
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378 | (mk_Sig … label ?)::compute_cost_trace_any_label … given_trace |
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379 | ] |
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380 | and compute_cost_trace_any_label |
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381 | (cost_labels: BitVectorTrie costlabel 16) |
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382 | (trace_ends_flag: trace_ends_with_ret) |
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383 | (start_status: Status) (final_status: Status) |
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384 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
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385 | on the_trace: list (Σk:costlabel. ∃pc. lookup_opt … pc cost_labels = Some … k) ≝ |
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386 | match the_trace with |
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387 | [ tal_base_not_return the_status _ _ _ _ ⇒ [] |
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388 | | tal_base_return the_status _ _ _ ⇒ [] |
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389 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
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390 | _ _ _ call_trace final_trace ⇒ |
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391 | let call_cost_trace ≝ compute_cost_trace_label_return … call_trace in |
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392 | let final_cost_trace ≝ compute_cost_trace_any_label cost_labels end_flag … final_trace in |
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393 | call_cost_trace @ final_cost_trace |
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394 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
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395 | compute_cost_trace_any_label cost_labels end_flag |
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396 | status_init status_end tail_trace |
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397 | ] |
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398 | and compute_cost_trace_label_return |
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399 | (cost_labels: BitVectorTrie costlabel 16) |
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400 | (start_status: Status) (final_status: Status) |
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401 | (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status) |
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402 | on the_trace: list (Σk:costlabel. ∃pc. lookup_opt … pc cost_labels = Some … k) ≝ |
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403 | match the_trace with |
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404 | [ tlr_base before after trace_to_lift ⇒ compute_cost_trace_label_label … trace_to_lift |
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405 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ |
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406 | let labelled_cost ≝ compute_cost_trace_label_label … labelled_trace in |
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407 | let return_cost ≝ compute_cost_trace_label_return … ret_trace in |
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408 | labelled_cost @ return_cost |
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409 | ]. |
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410 | [ %{pc} whd in match label; generalize in match labelled_proof; whd in ⊢ (% → ?); |
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411 | cases (lookup_opt costlabel … pc cost_labels) normalize |
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412 | [ #abs cases abs | // ] |
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413 | | // ] |
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414 | qed. |
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415 | |
---|
416 | (* XXX: zero in base cases (where the trace is a singleton transition) because |
---|
417 | we are counting until one from end |
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418 | *) |
---|
419 | let rec trace_any_label_length |
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420 | (cost_labels: BitVectorTrie costlabel 16) (trace_ends_flag: trace_ends_with_ret) |
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421 | (start_status: Status) (final_status: Status) |
---|
422 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
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423 | on the_trace: nat ≝ |
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424 | match the_trace with |
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425 | [ tal_base_not_return the_status _ _ _ _ ⇒ 0 |
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426 | | tal_base_return the_status _ _ _ ⇒ 0 |
---|
427 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final _ _ _ call_trace final_trace ⇒ |
---|
428 | let tail_length ≝ trace_any_label_length … final_trace in |
---|
429 | let pc_difference ≝ nat_of_bitvector … (program_counter … after_fun_call) - nat_of_bitvector … (program_counter … pre_fun_call) in |
---|
430 | pc_difference + tail_length |
---|
431 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
---|
432 | let tail_length ≝ trace_any_label_length … tail_trace in |
---|
433 | let pc_difference ≝ nat_of_bitvector … (program_counter … status_init) - nat_of_bitvector … (program_counter … status_pre) in |
---|
434 | pc_difference + tail_length |
---|
435 | ]. |
---|
436 | |
---|
437 | (* XXX: commented out in favour of above |
---|
438 | let rec trace_label_label_length |
---|
439 | (cost_labels: BitVectorTrie costlabel 16) (trace_ends_flag: trace_ends_with_ret) |
---|
440 | (start_status: Status) (final_status: Status) |
---|
441 | (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
---|
442 | on the_trace: nat ≝ |
---|
443 | match the_trace with |
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444 | [ tll_base ends_flag initial final given_trace labelled_proof ⇒ ? |
---|
445 | ] |
---|
446 | and trace_any_label_length |
---|
447 | (cost_labels: BitVectorTrie costlabel 16) (trace_ends_flag: trace_ends_with_ret) |
---|
448 | (start_status: Status) (final_status: Status) |
---|
449 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
---|
450 | on the_trace: nat ≝ |
---|
451 | match the_trace with |
---|
452 | [ tal_base_not_return the_status _ _ _ _ ⇒ ? |
---|
453 | | tal_base_return the_status _ _ _ ⇒ ? |
---|
454 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final _ _ _ call_trace final_trace ⇒ ? |
---|
455 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ ? |
---|
456 | ] |
---|
457 | and trace_label_return_length |
---|
458 | (cost_labels: BitVectorTrie costlabel 16) (start_status: Status) (final_status: Status) |
---|
459 | (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status) |
---|
460 | on the_trace: nat ≝ |
---|
461 | match the_trace with |
---|
462 | [ tlr_base before after trace_to_lift ⇒ ? |
---|
463 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ ? |
---|
464 | ]. |
---|
465 | cases daemon |
---|
466 | qed. |
---|
467 | *) |
---|
468 | |
---|
469 | (* Experiments with block_cost first! |
---|
470 | lemma reachable_program_counter_to_compute_trace_any_label_length_le_program_size: |
---|
471 | ∀start_status, final_status: Status. |
---|
472 | ∀code_memory: BitVectorTrie Byte 16. |
---|
473 | ∀program_size: nat. |
---|
474 | ∀cost_labels: BitVectorTrie costlabel 16. |
---|
475 | ∀trace_ends_flag: trace_ends_with_ret. |
---|
476 | ∀the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status. |
---|
477 | let start_program_counter ≝ program_counter … start_status in |
---|
478 | let final_program_counter ≝ program_counter … final_status in |
---|
479 | reachable_program_counter code_memory program_size start_program_counter → |
---|
480 | nat_of_bitvector … start_program_counter ≤ program_size + (nat_of_bitvector … final_program_counter). |
---|
481 | cases daemon |
---|
482 | qed. |
---|
483 | |
---|
484 | lemma reachable_program_counter_to_compute_trace_label_label_length_le_program_size: |
---|
485 | ∀start_status, final_status: Status. |
---|
486 | ∀code_memory: BitVectorTrie Byte 16. |
---|
487 | ∀program_size: nat. |
---|
488 | ∀cost_labels: BitVectorTrie costlabel 16. |
---|
489 | ∀trace_ends_flag: trace_ends_with_ret. |
---|
490 | ∀the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status. |
---|
491 | let start_program_counter ≝ program_counter … start_status in |
---|
492 | let final_program_counter ≝ program_counter … final_status in |
---|
493 | reachable_program_counter code_memory program_size start_program_counter → |
---|
494 | nat_of_bitvector … start_program_counter ≤ program_size + (nat_of_bitvector … final_program_counter). |
---|
495 | #start_status #final_status #code_memory #program_size #cost_labels #trace_ends_flag #the_trace |
---|
496 | cases daemon |
---|
497 | qed. |
---|
498 | |
---|
499 | corollary reachable_program_counter_to_compute_trace_length_le_program_size: |
---|
500 | ∀start_status, final_status: Status. |
---|
501 | ∀code_memory: BitVectorTrie Byte 16. |
---|
502 | ∀program_size: nat. |
---|
503 | ∀cost_labels: BitVectorTrie costlabel 16. |
---|
504 | ∀trace_ends_flag: trace_ends_with_ret. |
---|
505 | ∀the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status. |
---|
506 | let program_counter ≝ program_counter … start_status in |
---|
507 | reachable_program_counter code_memory program_size program_counter → |
---|
508 | trace_length start_status final_status ≤ program_size. |
---|
509 | cases daemon |
---|
510 | qed. *) |
---|
511 | |
---|
512 | include alias "arithmetics/nat.ma". |
---|
513 | include alias "basics/logic.ma". |
---|
514 | |
---|
515 | lemma and_intro_right: |
---|
516 | ∀a, b: Prop. |
---|
517 | a → (a → b) → a ∧ b. |
---|
518 | #a #b /3/ |
---|
519 | qed. |
---|
520 | |
---|
521 | lemma lt_m_n_to_exists_o_plus_m_n: |
---|
522 | ∀m, n: nat. |
---|
523 | m < n → ∃o: nat. m + o = n. |
---|
524 | #m #n |
---|
525 | cases m |
---|
526 | [1: |
---|
527 | #irrelevant |
---|
528 | %{n} % |
---|
529 | |2: |
---|
530 | #m' #lt_hyp |
---|
531 | %{(n - S m')} |
---|
532 | >commutative_plus in ⊢ (??%?); |
---|
533 | <plus_minus_m_m |
---|
534 | [1: |
---|
535 | % |
---|
536 | |2: |
---|
537 | @lt_S_to_lt |
---|
538 | assumption |
---|
539 | ] |
---|
540 | ] |
---|
541 | qed. |
---|
542 | |
---|
543 | lemma lt_O_n_to_S_pred_n_n: |
---|
544 | ∀n: nat. |
---|
545 | 0 < n → S (pred n) = n. |
---|
546 | #n |
---|
547 | cases n |
---|
548 | [1: |
---|
549 | #absurd |
---|
550 | cases(lt_to_not_zero 0 0 absurd) |
---|
551 | |2: |
---|
552 | #n' #lt_hyp % |
---|
553 | ] |
---|
554 | qed. |
---|
555 | |
---|
556 | lemma exists_plus_m_n_to_exists_Sn: |
---|
557 | ∀m, n: nat. |
---|
558 | m < n → ∃p: nat. S p = n. |
---|
559 | #m #n |
---|
560 | cases m |
---|
561 | [1: |
---|
562 | #lt_hyp %{(pred n)} |
---|
563 | @(lt_O_n_to_S_pred_n_n … lt_hyp) |
---|
564 | |2: |
---|
565 | #m' #lt_hyp %{(pred n)} |
---|
566 | @(lt_O_n_to_S_pred_n_n) |
---|
567 | @(transitive_le … (S m') …) |
---|
568 | [1: |
---|
569 | // |
---|
570 | |2: |
---|
571 | @lt_S_to_lt |
---|
572 | assumption |
---|
573 | ] |
---|
574 | ] |
---|
575 | qed. |
---|
576 | |
---|
577 | |
---|
578 | let rec block_cost_trace_any_label_static_dynamic_ok |
---|
579 | (total_program_size: nat) (cost_labels: BitVectorTrie costlabel 16) |
---|
580 | (trace_ends_flag: ?) (start_status: Status) (final_status: Status) |
---|
581 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
---|
582 | on the_trace: |
---|
583 | let code_memory ≝ code_memory … start_status in |
---|
584 | let program_counter ≝ program_counter … start_status in |
---|
585 | ∀reachable_program_counter_witness: reachable_program_counter code_memory total_program_size program_counter. |
---|
586 | ∀good_program_witness: good_program code_memory total_program_size. |
---|
587 | (∃n: nat. trace_any_label_length … the_trace + (S n) = total_program_size) ∧ |
---|
588 | block_cost code_memory program_counter total_program_size cost_labels reachable_program_counter_witness good_program_witness = |
---|
589 | compute_paid_trace_any_label cost_labels trace_ends_flag start_status final_status the_trace ≝ ?. |
---|
590 | letin code_memory ≝ (code_memory … start_status) |
---|
591 | letin program_counter ≝ (program_counter … start_status) |
---|
592 | #reachable_program_counter_witness #good_program_witness |
---|
593 | generalize in match (refl … final_status); |
---|
594 | generalize in match (refl … start_status); |
---|
595 | cases the_trace in ⊢ (???% → ???% → %); |
---|
596 | [1: |
---|
597 | #start_status' #final_status' #execute_hyp #classifier_hyp #costed_hyp |
---|
598 | #start_status_refl #final_status_refl destruct |
---|
599 | whd in match (trace_any_label_length ? ? ? ? ?); |
---|
600 | whd in match (compute_paid_trace_any_label ? ? ? ? ?); |
---|
601 | @and_intro_right |
---|
602 | [1: |
---|
603 | generalize in match reachable_program_counter_witness; |
---|
604 | whd in match (reachable_program_counter code_memory total_program_size program_counter); |
---|
605 | * #irrelevant #relevant |
---|
606 | lapply (exists_plus_m_n_to_exists_Sn (nat_of_bitvector … program_counter) total_program_size relevant) |
---|
607 | #hyp assumption |
---|
608 | |2: |
---|
609 | * #n #trace_length_hyp |
---|
610 | whd in match block_cost; normalize nodelta |
---|
611 | generalize in match reachable_program_counter_witness; |
---|
612 | generalize in match good_program_witness; |
---|
613 | <trace_length_hyp in ⊢ (∀_:?. ∀_:?. ??(???%??????)?); |
---|
614 | -reachable_program_counter_witness #reachable_program_counter_witness |
---|
615 | -good_program_witness #good_program_witness |
---|
616 | whd in ⊢ (??%?); @pair_elim |
---|
617 | ] |
---|
618 | |2: |
---|
619 | #start_status' #final_status' #execute_hyp #classified_hyp |
---|
620 | #start_status_refl #final_status_refl destruct |
---|
621 | whd in match (trace_any_label_length ? ? ? ? ?); |
---|
622 | whd in match (compute_paid_trace_any_label ? ? ? ? ?); |
---|
623 | @and_intro_right |
---|
624 | [1: |
---|
625 | generalize in match reachable_program_counter_witness; |
---|
626 | whd in match (reachable_program_counter code_memory total_program_size program_counter); |
---|
627 | * #irrelevant #relevant |
---|
628 | lapply (exists_plus_m_n_to_exists_Sn (nat_of_bitvector … program_counter) total_program_size relevant) |
---|
629 | #hyp assumption |
---|
630 | |2: |
---|
631 | * #n #trace_length_hyp |
---|
632 | whd in match block_cost; normalize nodelta |
---|
633 | generalize in match reachable_program_counter_witness; |
---|
634 | generalize in match good_program_witness; |
---|
635 | <trace_length_hyp in ⊢ (∀_:?. ∀_:?. ??(???%??????)?); |
---|
636 | -reachable_program_counter_witness #reachable_program_counter_witness |
---|
637 | -good_program_witness #good_program_witness |
---|
638 | whd in match (0 + S n); whd in ⊢ (??%?); |
---|
639 | ] |
---|
640 | |3: |
---|
641 | #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call |
---|
642 | #status_final #execute_hyp #classifier_hyp #after_return_hyp |
---|
643 | #trace_label_return #trace_any_label #start_status_refl #final_status_refl |
---|
644 | destruct |
---|
645 | whd in match (trace_any_label_length ? ? ? ? ?); |
---|
646 | whd in match (compute_paid_trace_any_label ? ? ? ? ?); |
---|
647 | @and_intro_right |
---|
648 | [1: |
---|
649 | |2: |
---|
650 | ] |
---|
651 | |4: |
---|
652 | #end_flag #status_pre #status_init #status_end #execute_hyp #trace_any_label |
---|
653 | #classifier_hyp #costed_hyp #start_status_refl #final_status_refl |
---|
654 | destruct |
---|
655 | whd in match (trace_any_label_length ? ? ? ? ?); |
---|
656 | whd in match (compute_paid_trace_any_label ? ? ? ? ?); |
---|
657 | @and_intro_right |
---|
658 | [1: |
---|
659 | |2: |
---|
660 | ] |
---|
661 | ] |
---|
662 | ] |
---|
663 | |
---|
664 | corollary block_cost_trace_label_labelstatic_dynamic_ok: |
---|
665 | ∀program_size: nat. |
---|
666 | ∀cost_labels: BitVectorTrie costlabel 16. |
---|
667 | ∀trace_ends_flag. |
---|
668 | ∀start_status: Status. |
---|
669 | ∀final_status: Status. |
---|
670 | ∀the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status. |
---|
671 | let code_memory ≝ code_memory … start_status in |
---|
672 | let program_counter ≝ program_counter … start_status in |
---|
673 | ∀reachable_program_counter_witness: reachable_program_counter code_memory program_size program_counter. |
---|
674 | ∀good_program_witness: good_program code_memory program_size. |
---|
675 | (∃n: nat. trace_length start_status final_status + n = program_size) ∧ |
---|
676 | block_cost code_memory program_counter program_size program_size cost_labels reachable_program_counter_witness good_program_witness ? = |
---|
677 | compute_paid_trace_label_label cost_labels trace_ends_flag start_status final_status the_trace. |
---|
678 | [2: |
---|
679 | @le_plus_n_r |
---|
680 | |1: |
---|
681 | #program_size #cost_labels #trace_ends_flag #start_status #final_status |
---|
682 | #the_trace normalize nodelta |
---|
683 | #reachable_program_counter_witness #good_program_witness |
---|
684 | generalize in match start_status; |
---|
685 | qed. |
---|
686 | |
---|
687 | (* |
---|
688 | (* To be moved elsewhere*) |
---|
689 | lemma le_S_n_m_to_le_n_m: ∀n,m. S n ≤ m → n ≤ m. |
---|
690 | #n #m #H change with (pred (S n) ≤ m) @(transitive_le ? (S n)) // |
---|
691 | qed. |
---|
692 | *) |
---|
693 | |
---|
694 | (* This shoudl go in bigops! *) |
---|
695 | theorem bigop_sum_rev: ∀k1,k2,p,B,nil.∀op:Aop B nil.∀f:nat→B. |
---|
696 | \big[op,nil]_{i<k1+k2|p i} (f i) = |
---|
697 | op \big[op,nil]_{i<k2|p (i+k1)} (f (i+k1)) \big[op,nil]_{i<k1|p i} (f i). |
---|
698 | #k1 #k2 #p #B #nil #op #f >bigop_sum >commutative_plus @same_bigop #i @leb_elim normalize |
---|
699 | [2,4: // |
---|
700 | | #H1 #H2 <plus_minus_m_m // |
---|
701 | | #H1 #H2 #H3 <plus_minus_m_m //] |
---|
702 | qed. |
---|
703 | |
---|
704 | (* This is taken by sigma_pi.ma that does not compile now *) |
---|
705 | definition natAop ≝ mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n)) |
---|
706 | (λa,b,c.sym_eq ??? (associative_plus a b c)). |
---|
707 | |
---|
708 | definition natACop ≝ mk_ACop nat 0 natAop commutative_plus. |
---|
709 | |
---|
710 | definition natDop ≝ mk_Dop nat 0 natACop times (λn.(sym_eq ??? (times_n_O n))) |
---|
711 | distributive_times_plus. |
---|
712 | |
---|
713 | unification hint 0 ≔ ; |
---|
714 | S ≟ mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n)) |
---|
715 | (λa,b,c.sym_eq ??? (associative_plus a b c)) |
---|
716 | (* ---------------------------------------- *) ⊢ |
---|
717 | plus ≡ op ? ? S. |
---|
718 | |
---|
719 | unification hint 0 ≔ ; |
---|
720 | S ≟ mk_ACop nat 0 (mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n)) |
---|
721 | (λa,b,c.sym_eq ??? (associative_plus a b c))) commutative_plus |
---|
722 | (* ---------------------------------------- *) ⊢ |
---|
723 | plus ≡ op ? ? S. |
---|
724 | |
---|
725 | unification hint 0 ≔ ; |
---|
726 | S ≟ natDop |
---|
727 | (* ---------------------------------------- *) ⊢ |
---|
728 | plus ≡ sum ? ? S. |
---|
729 | |
---|
730 | unification hint 0 ≔ ; |
---|
731 | S ≟ natDop |
---|
732 | (* ---------------------------------------- *) ⊢ |
---|
733 | times ≡ prod ? ? S. |
---|
734 | |
---|
735 | notation > "Σ_{ ident i < n } f" |
---|
736 | with precedence 20 |
---|
737 | for @{'bigop $n plus 0 (λ${ident i}.true) (λ${ident i}. $f)}. |
---|
738 | |
---|
739 | notation < "Σ_{ ident i < n } f" |
---|
740 | with precedence 20 |
---|
741 | for @{'bigop $n plus 0 (λ${ident i}:$X.true) (λ${ident i}:$Y. $f)}. |
---|
742 | |
---|
743 | axiom code_memory_ro_label_label: |
---|
744 | ∀cost_labels. ∀ends_flag. |
---|
745 | ∀initial,final. trace_label_label (ASM_abstract_status cost_labels) ends_flag initial final → |
---|
746 | code_memory … initial = code_memory … final. |
---|
747 | |
---|
748 | (* |
---|
749 | axiom code_memory_ro_label_return: |
---|
750 | ∀cost_labels. |
---|
751 | ∀initial,final. trace_label_return (ASM_abstract_status cost_labels) initial final → |
---|
752 | code_memory … initial = code_memory … final. |
---|
753 | *) |
---|
754 | |
---|
755 | definition tech_cost_of_label0: |
---|
756 | ∀cost_labels: BitVectorTrie costlabel 16. |
---|
757 | ∀cost_map: identifier_map CostTag nat. |
---|
758 | ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k). |
---|
759 | ∀ctrace:list (Σk:costlabel.∃pc. lookup_opt costlabel 16 pc cost_labels = Some ? k). |
---|
760 | ∀i,p. present … cost_map (nth_safe ? i ctrace p). |
---|
761 | #cost_labels #cost_map #codom_dom #ctrace #i #p |
---|
762 | cases (nth_safe … i ctrace ?) normalize #id * #id_pc #K |
---|
763 | lapply (codom_dom … K) #k_pres >(lookup_lookup_present … k_pres) |
---|
764 | % #abs destruct (abs) |
---|
765 | qed. |
---|
766 | |
---|
767 | include alias "arithmetics/nat.ma". |
---|
768 | include alias "basics/logic.ma". |
---|
769 | |
---|
770 | lemma ltb_rect: |
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771 | ∀P:Type[0].∀n,m. (n < m → P) → (¬ n < m → P) → P. |
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772 | #P #n #m lapply (refl … (ltb n m)) cases (ltb n m) in ⊢ (???% → %); #E #H1 #H2 |
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773 | [ @H1 @leb_true_to_le @E | @H2 @leb_false_to_not_le @E ] |
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774 | qed. |
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775 | |
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776 | lemma same_ltb_rect: |
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777 | ∀P,n,m,H1,H2,n',m',H1',H2'. |
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778 | ltb n m = ltb n' m' → (∀x,y. H1 x = H1' y) → (∀x,y. H2 x = H2' y) → |
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779 | ltb_rect P n m H1 H2 = |
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780 | ltb_rect P n' m' H1' H2'. |
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781 | #P #n #m #H1 #H2 #n' #m' #H1' #H2' #E #K1 #K2 whd in ⊢ (??%?); |
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782 | cut (∀xxx,yyy,xxx',yyy'. |
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783 | match ltb n m |
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784 | return λx:bool. |
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785 | eq bool (ltb n m) x |
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786 | → (lt n m → P) → (Not (lt n m) → P) → P |
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787 | with |
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788 | [ true ⇒ |
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789 | λE0:eq bool (ltb n m) true. |
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790 | λH10:lt n m → P. |
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791 | λH20:Not (lt n m) → P. H10 (xxx E0) |
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792 | | false ⇒ |
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793 | λE0:eq bool (ltb n m) false. |
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794 | λH10:lt n m → P. |
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795 | λH20:Not (lt n m) → P. H20 (yyy E0)] |
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796 | (refl … (ltb n m)) H1 H2 = |
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797 | match ltb n' m' |
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798 | return λx:bool. |
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799 | eq bool (ltb n' m') x |
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800 | → (lt n' m' → P) → (Not (lt n' m') → P) → P |
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801 | with |
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802 | [ true ⇒ |
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803 | λE0:eq bool (ltb n' m') true. |
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804 | λH10:lt n' m' → P. |
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805 | λH20:Not (lt n' m') → P. H10 (xxx' E0) |
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806 | | false ⇒ |
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807 | λE0:eq bool (ltb n' m') false. |
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808 | λH10:lt n' m' → P. |
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809 | λH20:Not (lt n' m') → P. H20 (yyy' E0)] |
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810 | (refl … (ltb n' m')) H1' H2' |
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811 | ) [2: #X @X] |
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812 | >E cases (ltb n' m') #xxx #yyy #xxx' #yyy' normalize |
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813 | [ @K1 | @K2 ] |
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814 | qed. |
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815 | |
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816 | |
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817 | definition tech_cost_of_label: |
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818 | ∀cost_labels: BitVectorTrie costlabel 16. |
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819 | ∀cost_map: identifier_map CostTag nat. |
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820 | ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k). |
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821 | list (Σk:costlabel.∃pc. lookup_opt costlabel 16 pc cost_labels = Some ? k) → |
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822 | nat → nat |
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823 | ≝ λcost_labels,cost_map,codom_dom,ctrace,i. |
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824 | ltb_rect ? i (|ctrace|) |
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825 | (λH. lookup_present ?? cost_map (nth_safe ? i ctrace H) ?) |
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826 | (λ_.0). |
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827 | @tech_cost_of_label0 @codom_dom |
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828 | qed. |
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829 | |
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830 | lemma shift_nth_safe: |
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831 | ∀T,i,l2,l1,K1,K2. |
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832 | nth_safe T i l1 K1 = nth_safe T (i+|l2|) (l2@l1) K2. |
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833 | #T #i #l2 elim l2 normalize |
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834 | [ #l1 #K1 <plus_n_O #K2 % |
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835 | | #hd #tl #IH #l1 #K1 <plus_n_Sm #K2 change with (? < ?) in K1; change with (? < ?) in K2; |
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836 | whd in ⊢ (???%); @IH ] |
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837 | qed. |
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838 | |
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839 | lemma tech_cost_of_label_shift: |
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840 | ∀cost_labels,cost_map,codom_dom,l1,l2,i. |
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841 | i < |l2| → |
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842 | tech_cost_of_label cost_labels cost_map codom_dom l2 i = |
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843 | tech_cost_of_label cost_labels cost_map codom_dom (l1@l2) (i+|l1|). |
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844 | #cost_labels #cost_Map #codom_dom #l1 #l2 #i #H |
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845 | whd in match tech_cost_of_label; normalize nodelta @same_ltb_rect |
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846 | [ @(ltb_rect ? i (|l2|)) @(ltb_rect ? (i+|l1|) (|l1@l2|)) #K1 #K2 |
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847 | whd in match ltb; normalize nodelta |
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848 | [1: >le_to_leb_true try assumption applyS le_to_leb_true // |
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849 | |4: >not_le_to_leb_false try assumption applyS not_le_to_leb_false |
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850 | change with (¬ ? ≤ ?) in K1; applyS K1 |
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851 | |2: @⊥ @(absurd (i+|l1| < |l1@l2|)) // >length_append |
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852 | applyS (monotonic_lt_plus_r … (|l1|)) // |
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853 | |3: @⊥ @(absurd ?? K2) >length_append in K1; #K1 /2 by lt_plus_to_lt_l/ ] |
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854 | | #H1 #H2 |
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855 | generalize in match (tech_cost_of_label0 ??? (l1@l2) ??); |
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856 | <(shift_nth_safe … H1) #p % |
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857 | | // ] |
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858 | qed. |
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859 | |
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860 | let rec compute_max_trace_label_return_cost_ok_with_trace |
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861 | (cost_labels: BitVectorTrie costlabel 16) |
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862 | (cost_map: identifier_map CostTag nat) |
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863 | (initial: Status) (final: Status) |
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864 | (trace: trace_label_return (ASM_abstract_status cost_labels) initial final) |
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865 | (codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k)) |
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866 | on trace: |
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867 | ∀dom_codom:(∀k. ∀k_pres:present … cost_map k. ∃pc. lookup_opt … pc cost_labels = Some … k → |
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868 | block_cost (code_memory … initial) cost_labels pc 2^16 = lookup_present … k_pres). |
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869 | let ctrace ≝ compute_cost_trace_label_return … trace in |
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870 | compute_max_trace_label_return_cost … trace = |
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871 | (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i)) |
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872 | ≝ ? |
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873 | and compute_max_trace_label_label_cost_ok_with_trace |
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874 | (cost_labels: BitVectorTrie costlabel 16) |
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875 | (trace_ends_flag: trace_ends_with_ret) |
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876 | (cost_map: identifier_map CostTag nat) |
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877 | (initial: Status) (final: Status) |
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878 | (trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag initial final) |
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879 | (codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k)) |
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880 | (dom_codom:(∀k. ∀k_pres:present … cost_map k. ∃pc. lookup_opt … pc cost_labels = Some … k → |
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881 | block_cost (code_memory … initial) cost_labels pc 2^16 = lookup_present … k_pres)) |
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882 | on trace: |
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883 | let ctrace ≝ compute_cost_trace_label_label … trace in |
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884 | compute_max_trace_label_label_cost … trace = |
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885 | (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i)) |
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886 | ≝ ?. |
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887 | cases trace |
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888 | [ #sb #sa #tr #dom_codom |
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889 | @compute_max_trace_label_label_cost_ok_with_trace @dom_codom |
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890 | | #si #sl #sf #tr1 #tr2 #dom_codom whd in ⊢ (let ctrace ≝ % in ??%?); |
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891 | change with (let ctrace ≝ ? in ? = bigop (|? @ ?|) ?????) |
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892 | >append_length in ⊢ (let ctrace ≝ ? in ???(?%?????)); |
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893 | change with (?=?) >bigop_sum_rev >commutative_plus @eq_f2 |
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894 | [ >(compute_max_trace_label_return_cost_ok_with_trace … cost_map … codom_dom) |
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895 | -compute_max_trace_label_return_cost_ok_with_trace |
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896 | [2:lapply (code_memory_ro_label_label … tr1) #E2 <E2 @dom_codom] |
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897 | @same_bigop [//] #i #H #_ -dom_codom @tech_cost_of_label_shift // |
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898 | | >(compute_max_trace_label_label_cost_ok_with_trace … cost_map … codom_dom … dom_codom) |
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899 | @same_bigop [//] #i #H #_ |
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900 | ] |
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901 | (* |
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902 | lemma |
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903 | compute_max_trace_any_label_cost cost_label trace_ends_flag start_status |
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904 | final_status the_trace = |
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905 | |
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906 | |
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907 | let rec compute_paid_trace_label_label_cost |
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908 | (cost_labels: BitVectorTrie Byte 16) |
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909 | (trace_ends_flag: trace_ends_with_ret) |
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910 | (start_status: Status) (final_status: Status) |
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911 | (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag |
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912 | start_status final_status) on the_trace: nat ≝ |
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913 | match the_trace with |
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914 | [ tll_base _ _ _ given_trace _ ⇒ |
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915 | compute_paid_trace_any_label_cost … given_trace |
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916 | ] |
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917 | and compute_paid_trace_any_label_cost |
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918 | (cost_labels: BitVectorTrie Byte 16) |
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919 | (trace_ends_flag: trace_ends_with_ret) |
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920 | (start_status: Status) (final_status: Status) |
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921 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
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922 | on the_trace: nat ≝ |
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923 | match the_trace with |
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924 | [ tal_base_not_return the_status _ _ _ _ ⇒ current_instruction_cost the_status |
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925 | | tal_base_return the_status _ _ _ ⇒ current_instruction_cost the_status |
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926 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
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927 | _ _ _ call_trace final_trace ⇒ |
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928 | let current_instruction_cost ≝ current_instruction_cost pre_fun_call in |
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929 | let final_trace_cost ≝ compute_paid_trace_any_label_cost cost_labels end_flag … final_trace in |
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930 | current_instruction_cost + final_trace_cost |
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931 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
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932 | let current_instruction_cost ≝ current_instruction_cost status_pre in |
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933 | let tail_trace_cost ≝ |
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934 | compute_paid_trace_any_label_cost cost_labels end_flag |
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935 | status_init status_end tail_trace |
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936 | in |
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937 | current_instruction_cost + tail_trace_cost |
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938 | ] |
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939 | and compute_paid_trace_label_return_cost |
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940 | (cost_labels: BitVectorTrie Byte 16) |
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941 | (start_status: Status) (final_status: Status) |
---|
942 | (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status) |
---|
943 | on the_trace: nat ≝ |
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944 | match the_trace with |
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945 | [ tlr_base before after trace_to_lift ⇒ compute_paid_trace_label_label_cost … trace_to_lift |
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946 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ |
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947 | let labelled_cost ≝ compute_paid_trace_label_label_cost … labelled_trace in |
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948 | let return_cost ≝ compute_paid_trace_label_return_cost … ret_trace in |
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949 | labelled_cost + return_cost |
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950 | ]. |
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951 | |
---|
952 | let rec trace_lab_rec_cost' (p: trace_lab_ret) : nat. |
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953 | | (call b bf tr af tl) as self ⇒ |
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954 | trace_lab_lab_cost_nocall self + trace_lab_ret_cost' tr + |
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955 | trace_lab_rec_cost' tl |
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956 | |
---|
957 | theorem main_lemma: |
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958 | ∀p. trace_lab_rec_cost p = trace_lab_rec_cost' p. |
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959 | |
---|
960 | axiom lemma1: |
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961 | ∀p: simple_path. |
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962 | traverse_cost_internal (pc (hd p)) … = trace_lab_lab_cost_nocall p. |
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963 | |
---|
964 | axiom lemma2: |
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965 | ∀s,l,cost_map. |
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966 | is_labelled l s → |
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967 | traverse_cost_internal s = cost_map l. |
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968 | |
---|
969 | axiom main_statement: |
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970 | ∀s. |
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971 | ∀cost_map. |
---|
972 | let p ≝ compute_simple_path0 s in |
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973 | ∑ (cost_trace p) cost_map = trace_lab_rec_cost' p. |
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974 | |
---|
975 | axiom main_statement: |
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976 | ∀s. |
---|
977 | ∀cost_map. |
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978 | let p ≝ compute_simple_path0 s in |
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979 | execute' (path_length p) s = 〈s',τ〉 → |
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980 | Timer s' - Timer s = ∑ τ cost_map. *) |
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